
\section{Triangulation metrics}
\label{sec:heuristics}

All minimality metrics are defined in the package \code{Triangulation.minimalityMetric} and implement the interface \code{MinimalityMetric}. Metrics that assign a weight to each triangle and minimize the sum of all triangle weights also implement the interface \code{MinimumTriangleSumMetric}. The following metrics are implemented:
\begin{itemize}
\itm{Shortest edges:} This metric minimizes the sum of all edge lengths of the triangulation. The diamond property can be used to speed up the triangulation process.
\itm{Longest base to height ratio:} This metric favours triangles where the length of the longest edge and the length of the height standing on this edge are similar. Like the shortest edge metric it minimizes the sum of the weight of all triangles that are part of the triangulation. The weight of a triangle is $r(l, h) = max(l, h) / min(l, h)$, where $l$ is the length of the longest edge of the triangle and $h$ the length of the height standing on it.\\ It is unknown whether the diamond property can be used with this metric, because of this the metric is implemented twice, with diamond metric and without (very slow).
\itm{Balanced angles:} The weight function is the ratio defined by the largest angle divided by the smallest angle of a triangle. The sum of all ratios of the triangulation is minimized. \cite{Dai2000} It is unknown whether the diamond property is applicable\footnote{Experiments suggest that the diamond property may be applicable with an angle $\leq\sfrac{\pi}{4}$\ .}.
\itm{Delaunay:} This metric maximizes the minimum inner angle of all triangles that are part of the triangulation. Since any optimal triangulation is locally optimal with regards to each pair of triangles sharing an edge, this heuristic works with the LMT triangulation algorithm. Unfortunately the polygon triangulation algorithm has not been adapted to work with this heuristic\footnote{Polygon holes are very rare in the LMT skeleton of uniformly distributed random point sets when the delaunay heuristic is used. We have not seen any in our experiments.}.
\end{itemize}
